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Unformatted text preview: 1.2 Inﬁnite Sets 21 Example 1.2.2. Suppose A = {1, 2, 3, 4, 5}. Then the subset {2, 3, 5} corresponds to
(0, 1, 1, 0, 1) ∈ 2A . This is the function 1 → 0,
1.2.3 Exercises: #1,3 2 → 1, 3 → 1, Recommended: #2,4 4 → 0, 5→1 Due: Jan. 29 1. Every subset of N is either ﬁnite or countable.
2. If A1 , A2 , A3 , . . . An are countable then n
k=1 Ak is countable. 3. Show that the set of algebraic numbers is countable. A number x is algebraic iﬀ
a0 + a1 x + a2 x2 + · · · + an xn = 0,
for some integers ai . Hint: for N ∈ N, there are only ﬁnitely many equations with
n + a0  + · · · + an  = N. 4. Is the set of all irrational real numbers countable? 22 Math 413 1.3 Preliminaries The rational numbers The evolution of numbers ...
N = {1, 2, 3, 4 . . . }.
To solve equations like x + 5 = 2, need to add negatives:
Z = {. . . , −2, −1, 0, 1, 2, . . . }.
To solve equations like x × 5 = 2, need to add rationals:
Q={ m.
.
. m, n ∈ Z, n = 0}.
n To solve equations like xn = 2, need to add roots like √
n 2. What if we add solutions to n all polynomials an x + . . . a1 x + a0 = 0? We get A with
π ∈ A,
/ but √ −1 ∈ A . So what is R anyway? Roughly:
.
.
R ≈ {lim xn . {xn } ⊆ Q, {xn } converges}. Problem: how to deﬁne lim xn only in terms of Q? The usual deﬁnition of limit says that
lim xn = L iﬀ
∀ε > 0, ∃N, n ≥ N =⇒ xn − L < ε. This is circular: cannot use a real number L to deﬁne itself. Cauchy sequences will overcome this. 1.3.1 The abstract structure of Q. Deﬁnition 1.3.1. A group is a set with an associative binary operation, an identity, and
inverses. Written additively, (G, +, 0) must satisfy
1. x, y ∈ G =⇒ x + y is a welldeﬁned element of G.
2. ∃!0 such that x + 0 = 0 + x = x, ∀x ∈ G. ...
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 Fall '11
 Wong
 Sets, Rational number, lim xn

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